Multiple Wada basins with common boundaries in nonlinear driven oscillators

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• 作者：Yongxiang Zhang (1) (2)
  Huaguang Zhang (1) (3)
  Wenzhong Gao (4)
  
  1. School of Information Science and Engineering ; Northeastern University ; Shenyang ; 110819 ; China
  2. College of Science ; Shenyang Agricultural University ; Shenyang ; 110866 ; China
  3. State Key Laboratory of Synthetical Automation for Process Industries ; Shenyang ; Liaoning ; People Republic of China
  4. Department of Electrical and Computer Engineering ; University of Denver ; 2390 S York St ; Denver ; CO ; 80208 ; USA
  
• 关键词：Wada basins ; Fractal basins ; Attractors ; Shallow arch ; Unpredictability
• 刊名：Nonlinear Dynamics
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：79
• 期：4
• 页码：2667-2674
• 全文大小：1,830 KB
• 参考文献：1. Li, GX, Moon, FC (1990) Fractal basin boundaries in a two-degree-of-freedom nonlinear system. Nonlinear Dyn. 1: pp. 209-219 CrossRef
  2. Hong, L, Xu, JX (2003) Chaotic saddles in Wada basin boundaries and their bifurcations by generalized cell mapping digraph (GCMD) method. Nonlinear Dyn. 32: pp. 371-385 CrossRef
  3. Thompson, JMT, Soliman, MS (1990) Fractal control boundaries of driven oscillators and their relevance to safe engineering design. Proc. R. Soc. Lond. A 428: pp. 1-13 CrossRef
  4. Eschenazi, E, Solari, HG, Gilmore, R (1989) Basins of attraction in driven dynamical systems. Phys. Rev. A 39: pp. 2609-2627 CrossRef
  5. Ueda, Y, Yoshida, S, Stewart, HB, Thompson, JMT (1990) Basin explosions and escape phenomena in the two-well duffing oscillator: compound global bifurcations organizing behavior. Philos. Trans. R. Soc. Lond. A 332: pp. 169-182 CrossRef
  6. Kennedy, J., Yorke, J.A.: Basin of Wada. Phys. D 51, 213鈥?25 (1991)
  7. Nusse, HE, Yorke, JA (1997) Dynamics: Numerical Explorations. Springer, New York
  8. Aguirre, J, Viana, RL, Sanju谩n, MAF (2009) Fractal structures in nonlinear dynamics. Rev. Mod. Phys. 81: pp. 333-386 CrossRef
  9. Sweet, D, Ott, E, Yorke, JA (1999) Topology in chaotic scattering. Nature 399: pp. 315-316 CrossRef
  10. Ott, E.: Chaos in Dynamical System. Cambridge University Press, Cambridge (2002)
  11. Viana, RL, Silva, EC, Kroetz, T, Caldas, IL, Roberto, M, Sanju谩n, MAF (2011) Fractal structures in nonlinear plasma physics. Philos. Trans. R. Soc. A 369: pp. 371-395 CrossRef
  12. Zhang, Y (2013) Strange nonchaotic attractors with Wada basins. Phys. D 259: pp. 26-36 CrossRef
  13. Seoane, JM, Sanju谩n, MAF (2013) New developments in classical chaotic scattering. Rep. Prog. Phys. 76: pp. 016001 CrossRef
  14. Aguirre, J, Sanju谩n, MAF (2002) Unpredictable behavior in the duffing oscillator: Wada basins. Phys. D 171: pp. 41-51 CrossRef
  15. Nusse, HE, Ott, E, Yorke, JA (1995) Saddle-node bifurcations on fractal basin boundaries. Phys. Rev. Lett. 75: pp. 2482-2485 CrossRef
  16. Poon, L, Campos, J, Ott, E, Grebogi, C (1996) Wada basin boundaries in chaotic scattering. Int. J. Bifurcat. Chaos 6: pp. 251-265 CrossRef
  17. Aguirre, J, Vallejo, JC, Sanju谩n, MAF (2001) Wada basins and chaotic invariant sets in the H茅non鈥揌eiles system. Phys. Rev. E 64: pp. 066208 CrossRef
  18. Aguirre, J, Sanju谩n, MAF (2003) Limit of small exits in open Hamiltonian systems. Phys. Rev. E 67: pp. 056201 CrossRef
  19. Thompson, JMT, Hunt, GW (1973) A General Theory of Elastic Stability. Wiley, London
  20. Cao, Q, Wiercigroch, M, Pavlovskaia, EE, Grebogi, C, Thompson, JMT (2006) Archetypal oscillator for smooth and discontinuous dynamics. Phys. Rev. E 74: pp. 046218 CrossRef
  21. Zhang, Y, Luo, G, Cao, Q, Lin, M (2014) Wada basin dynamics of a shallow arch oscillator with more than 20 coexisting low-period periodic attractors. Int. J. Nonlinear Mech. 58: pp. 151-161 CrossRef
  22. Zhang, Y, Lu, LF (2014) Basin boundaries with nested structure in a shallow arch oscillator. Nonlinear Dyn. 77: pp. 1121-1132 CrossRef
  23. Nusse, HE, Yorke, JA (1996) Wada basin boundaries and basin cells. Phys. D 90: pp. 242-261 CrossRef
  24. Nusse, HE, Yorke, JA (1996) Basin of attraction. Science 271: pp. 1376-1380 CrossRef
  25. Zhang, Y (2013) Switching-induced Wada basin boundaries in the H茅non map. Nonlinear Dyn. 73: pp. 2221-2229 CrossRef
  26. Nusse, HE, Yorke, JA (2000) Fractal basin boundaries generated by basin cells and the geometry of mixing chaotic flows. Phys. Rev. Lett. 84: pp. 626-629 CrossRef
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

In driven oscillators, examples are shown that every boundary point of one basin is on the boundary of the two remaining basins and all three boundaries of these basins coincide. When there are more than five basins of attraction, is it possible that every boundary point of one basin is on the boundary of other basins? Is it possible that all basin boundaries coincide? This paper describes some numerical experiments giving evidence of seven Wada basin boundaries and all seven basin boundaries coincide for a driven shallow arch oscillator. The results are verified by the basin cell theory. It suggests that small noises can lead to final state sensitivity and uncertain dynamics in the driven oscillator.